Time-Domain Electromagnetic Analysis : From FDTD to Wavelets and Beyond

Research supported by the U.S. Army Research Office and the U.S. Army CECOM

-23.7dB

-10.3dB

C o s t a s D. S a r r i s

The Radiation Laboratory

Department of Electrical Engineering

and Computer Science

The University of Michigan, Ann Arbor

Advisor : Prof. Linda P. B. Katehi

The University of Toronto, June 28, 2002

Time-Domain Time-Domain ElectromagneticElectromagnetic

Analysis : Analysis :

From FDTD to From FDTD to Wavelets and BeyondWavelets and Beyond

The University of Toronto, Toronto, Ontario, Canada, June 28, 2002

Outline

Introduction

• Time Domain numerical schemes for Maxwell’s equations : Research motivation and state-of-the art.• Wavelet based numerical schemes : How they started, where they are.• The Multi-Resolution Time-Domain technique.

Recent developments in wavelet-based schemes

• Formulation and dispersion analysis.• Numerical interface between FDTD and MRTD : Efficient implementation of boundary and Perfectly Matched Layer conditions.• Implementation of dynamic mesh adaptivity : The case study of a nonlinear optical pulse propagation.

Advanced Application : Wireless channel modeling

• Cosite Interference in VHF transceiver networks.• Mixed electromagnetic-circuit simulations.• Modeling of cosite interference in a forest environment.

Trends and Conclusions


Example : FDTD discretization of

y

E

z

E1

t

H zyx

y

EE

z

EEtHH

z2/1k,j,in

z2/1k,1j,in

yk,2/1j,in

y1k,2/1j,inx

2/1k,2/1j,i2/1nx

2/1k,2/1j,i2/1n

y

EE

z

EEtHH

z2/1k,j,in

z2/1k,1j,in

yk,2/1j,in

y1k,2/1j,inx

2/1k,2/1j,i2/1nx

2/1k,2/1j,i2/1n

Marching in time scheme

• In FDTD (K.S. Yee, 1966), the computational domain is divided in Yee’s cells and Maxwell’s equations are solved by marching in time.

Finite Difference – Time Domain (FDTD) Method

Simple, versatile and robust algorithm.

Second order accurate in space and

time.

Inherently parallelizable.

Ten-twenty points per wavelength

necessary.

Small time step necessary.

Multi-wavelength domains result to

extremely large scale problems.


• In MRTD [Krumpholz and Katehi, MTT-T 1996], field components are expanded in a wavelet basis :

)kξ2(ψd)k(ξφcξf j

k jk,j

kk

Coarse approximation

Successive wavelet refinement

=+

Multiresolution Decomposition

Example : Mesh Refinement in Haar wavelet MRTD

Use of N wavelet levels in

direction , brings about a mesh

refinement by a factor :

N2

Multi-Resolution - Time Domain (MRTD) Method

Wavelet coefficients are significant near field variations/discontinuities.

MRTD offers a natural framework for the implementation of an adaptive, moving mesh.


Comparison of Novel Time Domain Schemes

Coarse discretization

Modeling of geometric

details

Modeling of geometric

details

Dynamic adaptivityDynamic adaptivity

Computational Efficiency Computational Efficiency

FDTD + subgridding

Subgrid at details +Interpolations/extrapolations

High Order Methods

Wavelet Based Methods

Higher accuracy /Larger stencil

Higher order Wavelet basis

Thresholding


Example : Haar Basis

Finite Domain functions – only nearest neighbor coupling of basis functions ( no stencil

effects ).

Extension of the scheme to higher orders is relatively simple.

Haar MRTD with scaling functions only is the FDTD scheme.

Low approximation order scheme ( FDTD and Haar MRTD need same number of

degrees of freedom).

Haar mother wavelet Haar mother wavelet Haar scaling function Haar scaling function

-0.25 0 0.25 0.5 0.75 1 1.250

0.25

0.5

0.75

1

-0.25 0 0.25 0.5 0.75 1 1.25-1.25

-0.75

-0.25

0.25

0.75

1.25


12,,1,0p

p)k)(ξ2ψ(2ξψr

r2r/rpk,

12,,1,0p

p)k)(ξ2ψ(2ξψr

r2r/rpk,

r=0

r=1

r=2

K-th cell

p=0 p=1 p=2 p=3

p=0 p=1

p=0

Haar scaling basis :

Higher order Haar wavelet functions are produced with translations and dilations of the mother wavelet.

Example : Order r wavelet in

k-th cell :

kk

Example : Haar Basis (cont-d)

Haar wavelets of orders

0, 1, 2


Battle-Lemarie scaling function Battle-Lemarie scaling function Battle-Lemarie mother wavelet Battle-Lemarie mother wavelet

-10 -5 0 5 10-0.2

0.2

0.6

1

-10 -5 0 5 10-1

-0.5

0

0.5

1

1.5

Example : Battle-Lemarie basis


Use of Wavelets for Mesh Refinement : Promise

Introduction of one wavelet level refines the mesh by a factor of two with respect to a scaling function based numerical scheme.

Example : Pulses + zero order Haar wavelets = FDTD scheme of half its original cell size per direction.

If is the scaling cell size, the effective cell size for an R-order MRTD is = 2 R+1.


Use of Wavelets for Mesh Refinement : Results

Battle-Lemarie wavelet based W-MRTD [Krumpholz and Katehi, MTT-T, April 1996] :

-Spurious modes observed, attributed to wavelets.

-Use of wavelets caused only incremental improvement in accuracy.

Battle-Lemarie wavelet based W-MRTD [Krumpholz and Katehi, MTT-T, April 1996] :

-Spurious modes observed, attributed to wavelets.

-Use of wavelets caused only incremental improvement in accuracy.

Haar wavelet based MRTD [Goverdhanam et al., Fujii and Hoefer] :

- For zero order MRTD, scaling and wavelet terms are given by uncoupled equations.

- Incremental accuracy improvement due to wavelets :

“..addition of one resolution of wavelets within a Haar based MRTD framework does not improve significantly the numerical accuracy of the underlying coarse grid scheme.”

Grivet-Talocia, IEEE MGWL, October 2000.

Haar wavelet based MRTD [Goverdhanam et al., Fujii and Hoefer] :

- For zero order MRTD, scaling and wavelet terms are given by uncoupled equations.

- Incremental accuracy improvement due to wavelets :

“..addition of one resolution of wavelets within a Haar based MRTD framework does not improve significantly the numerical accuracy of the underlying coarse grid scheme.”

Grivet-Talocia, IEEE MGWL, October 2000.

These observations are in stark contradiction with Multiresolution Analysis principles, as established in approximation theory and signal processing.


Dispersion Analysis

Dispersion analysis consists in the determination of the numerical wavenumber as a function of the numerical frequency :

In practical terms, it answers the question of how coarse a grid can become for an allowed level of phase errors (important for CAD oriented algorithms).

FDTD dispersion analysis [Taflove, 1994] :

Substitute all terms in finite difference equations with plane wave type expressions,

Formulate a linear homogeneous system with respect to the amplitudes , impose condition that the system have a non-trivial solution.

numnum kk numnum kk

tnizmyqxpkim,q,pn ee

~ tnizmyqxpkim,q,pn ee

~

H,E p, q , m : space cell indices

n : time step index

x, y, z

Not directly applicable to MRTD, due to the multi-level character of its expansion basis.


Dispersion Analysis has been re-formulated from first principles of Fourier analysis.

1. In the MRTD finite difference equations, replace all scaling coefficients with FDTD type plane wave expressions (one dimension) :

2. Observe by simple Fourier calculus arguments that :

3. Formulate a linear system with respect to scaling spectral amplitudes ONLY.

Note : The number of unknowns of the system is independent of MRTD order.

A Modified Fourier Dispersion Analysis Method for MRTD1

[1] : C. D. Sarris, L. P. B. Katehi, “Some Aspects of Dispersion Analysis of MRTD Schemes”, 2001 ACES Conference.

Three step approachThree step approach

tnixqki,,qn ee

~ tnixqki,,qn ee

~

y, z

,qn

,qn

p,rp,r ,qn

,qn

p,rp,r

xkˆ

2/xkˆe2xk

r2/ipX2/r r

p,r

xkˆ

2/xkˆe2xk

r2/ipX2/r r

p,r


Dispersion Analysis : Gridding Effects

Most MRTD studies in the literature so far have been using a standard half scaling cell offset between electric and magnetic field nodes ( referred to from now on as Formulation IFormulation I ).

: Electric field scaling function

: Magnetic field scaling function

Haar MRTD Battle-Lemarie MRTD

: Electric field scaling node

: Magnetic field scaling node


Dispersion Analysis : Effect of Gridding on Haar MRTD

Zero order MRTD dispersion performance coincides with FDTD of cell size equal to the

scaling cell size.

In general, this formulation of Haar MRTD schemes, leads to a decrease in resolution by a

factor of two in ALL orders.

• Dispersion analyses of arbitrary

order MRTD schemes were

performed, based on the modified

Fourier method for MRTD.

• One Dimensional Propagation

results are shown.

• Notation :

xkX xkX Normalized Wavenumber

t t Normalized Frequency


Gridding Effects (cont-d)

• A better approach : Vary the offset between electric and magnetic field nodes so that it remain half

the MRTD equivalent cell, taking into account the mesh refinement brought about by the wavelets :

2rmax2

1

2

1s

• Equivalent grid points in 0-

Haar MRTD are determined

under the new convention :

s=0.25 (rmax = 0).

• The equivalent grid

points are properly offset.

Formulation II (proposed in Formulation II (proposed in this work)this work)

Reference : Sarris and Katehi, IEEE MTT-T, Dec. 2001


Dispersion Analysis : Effect of Gridding on Haar MRTD (ii)

Each wavelet level increases the Nyquist limit of the scheme (turning point in the above

curves) by a factor of 2 (FDTD Nyquist point is X / = 1 ) => Consistence with MRA has been

enforced.

• Dispersion analysis of arbitrary order MRTD schemes under Formulation II has been performed.

• Results for one dimensional wave propagation are shown.


Dispersion Analysis : Effect of Gridding on W-MRTD

Applying the same gridding principles, the W-MRTD scheme [Krumpholz and Katehi, 1996] is reformulated.

Formulation 2 : Quarter cell E/H offset

Formulation 1 : Half cell E/H offset

Nyquist limit of the reformulated W-MRTD scheme is now at 2 .


Results : Resonant Frequencies for a two-dimensional air-filled cavity

(z)

(x)

a=32 cm

a

• TEz modes have been numerically determined for a square air-filled cavity, using FDTD and MRTD according to the two formulations under comparison.

• In all MRTD cases 32 by 32 degrees of freedom have been used.

• Results from FDTD of 32 by 32 and 16 by 16 cells are used for comparison.

Electric field Ey distribution for TE22, TE32 determined by 4 by 4 Haar MRTD (1 by 1 scaling mesh).


Resonant Frequencies for a two-dimensional air-filled cavity (cont-d)

(n,m) Fn,m

[GHz]

W-MRTD(form. I)

Rel. Error (%)

W-MRTD(form. II)

Rel. Error (%)

(1, 1) 0.6625 0.6620 - 0.0755 0.6627 + 0.0302

(2, 1) 1.0474 1.0451 - 0.2196 1.0466 - 0.0764

(2, 2) 1.3249 1.3233 - 0.1208 1.3240 - 0.0679

(3, 1) 1.4813 1.4714 - 0.6683 1.4793 - 0.1350

(3, 2) 1.6889 1.6827 - 0.3671 1.6878 - 0.0651

(3, 3) 1.9874 1.9818 - 0.2818 1.9853 - 0.1057

• Simulations for the schemes under comparison were run at a time step corresponding to 0.9 of their CFL number.

• MRTD schemes of various orders up to 4 per direction (scaling cell size = 32 cm) were tested.

• Haar MRTD (Form. I) follows a 16x16 FDTD in accuracy.

• Haar MRTD (Form. II) consistently follows a 32x32 FDTD.


W-MRTD : One Dimensional Case Study Example : One dimensional cavity structure.

Domain defined by 3+2 Battle-Lemarie scaling functions.Domain defined by 3+2 Battle-Lemarie scaling functions.

]GHz[n75.3fn

• Theoretical Dispersion Relationship :

]GHz[n25.0Xn

• A Battle-Lemarie scaling and a wavelet function are used as initial data (injecting normalized wavenumbers from 0 to 2 ), exciting the first 7 modes.

• W-MRTD schemes of formulations I, II are employed.

• Both schemes are run at half their CFL number.


W-MRTD I, II : Field Spectrum

• The spectrum of W-MRTD (form. I) contains a mode that appears to be spurious (but..it is not [Sarris and Katehi, IEEE MGWL Feb. 2001] ! ) .

• W-MRTD (form. II) appears to resolve all seven modes.


• Approach of this work : Combine the versatility of FDTD with the adaptivity of MRTD, by developing a hybrid FDTD / MRTD scheme based on a numerical interface .

In typical microwave circuit geometries..

In the modeling of the PML absorber ( hence in all kinds of open problems ).

An efficient solution to such problems is essential for the application of MRTD to practical microwave

structures.

Desired (coarse) MRTD mesh

Hybridization of MRTD : Motivation

Reference : Sarris and Katehi, Proc. 2001 IEEE IMS, paper submitted to MTT-T, April 2002.


Matching of MRTD / FDTD dispersion properties at the interface

1250 1300 1350 1400 1450 1500 1550 16000

0.5

1

1250 1300 1350 1400 1450 1500 1550 16000

0.5

1

1250 1300 1350 1400 1450 1500 1550 16000

0.5

1

Space Cell

4th order MRTD FDTD

1250 1300 1350 1400 1450 1500 1550 16000

0.5

1

1250 1300 1350 1400 1450 1500 1550 16000

0.5

1

1250 1300 1350 1400 1450 1500 1550 16000

0.5

1

Space Cell

4th order MRTD FDTD

2

effz

eff

2

effx

eff

2

p 2

zksin

z

1

2

xksin

x

1

2

tsin

tu

1

Haar MRTD

FDTD 2

FDz

FD

2

FDx

FD

2

p 2

zksin

z

1

2

xksin

x

1

2

tsin

tu

1

Evidently, letting :

or :

the dispersion properties of the two schemes match.

FDeff xx FDeff zz

A reflectionless propagation through an FDTD / Haar MRTD interface can be simulated

without interpolations / extrapolations.

FDTD cell size =

MRTD effective cell size


Two dimensional FDTD / MRTD Interface

• Overlapping of the two regions and use of field equivalence principles to de-couple the two

problems (FDTD and MRTD).

• Vehicle for data transfer between the two regions is the Fast Wavelet Transform, at the

optimal cost O(N).


FDTD / MRTD interface : Validation

• 3 by 4 MRTD interfaced with FDTD (MRTD mesh = 1x1).

• Effective cell size : 1cm x 1cm.

• The first six cavity modes are extracted via FDTD and the interface.


FDTD / MRTD interface : Mode Patterns

MRTD mesh 1x1 (order 4 by 4)

MRTD mesh 4x2 (order 2 by 3)

• TE21 and TE22 are extracted via the FDTD / MRTD interface.

• Absolutely smooth mode patterns are observed.


FDTD / MRTD interface : Metal fin loaded cavity

Geometry and Mesh for the FDTD / 2 x 2 MRTD interface

The pure MRTD modeling of this structure is restricted by the size of the scaling

function.

When this size exceeds the width/height of the fin, the fin scaling cell

unphysically couples the regions above and below the fin. Domain split is then

necessary.


Metal fin loaded cavity : Results

A Gaussian excitation with its 3 – dB bandwidth equal to half the cut-off frequency of the TE11 mode of the cavity is used.

Absolutely stable performance of the code is observed (electric field sampled in the cavity for time steps 18,000-20,000 is shown).


MRTD / FDTD PML interface

A common problem of wavelet based numerical methods is that they need many grid points to model boundary conditions.

For MRTD, it may seem necessary that at least the degrees of freedom of one MRTD cell are used to model the PML region.

However, since PML itself is terminated into a PEC, the following mesh configuration can be used to terminate the MRTD mesh in a PML region that extends over a fraction of the grid points of a single MRTD cell :

The termination of a second order MRTD mesh in a four grid point PML is assumed.

One MRTD cell contains 8 equivalent grid points.

All grid points that are beyond the PML region are zeroed out and used in the fast wavelet transform of the interface as such.

The termination of a second order MRTD mesh in a four grid point PML is assumed.

One MRTD cell contains 8 equivalent grid points.

All grid points that are beyond the PML region are zeroed out and used in the fast wavelet transform of the interface as such.

: 2nd order MRTD equivalent grid points

: FDTD grid points


MRTD / FDTD PML interface : Dipole Radiation

Mesh in FDTD cells

• An MRTD mesh is terminated in an FDTD based Uniaxial PML absorber ( 8 cells, theoretical reflection coefficient R = exp(-16), order 4 polynomial variation of conductivity) .

• The problem of an infinitesimal y-axis directed current element radiation is simulated via the FDTD / MRTD interface.

• Results are compared to FDTD.

0t/t40

30y et/tt/t4

zx

1tJ

00 f4

1t

GHz1f0

mm5.2zx

secp5.4t


Dipole Radiation : Results

0 by 0 order MRTD (32x32 cells) 2 by 2 order MRTD (8x8 cells) 4 by 4 order MRTD (2x2 cells)

• MRTD results are compared to FDTD for three different schemes.

• Field sampling points are at A, B, C respectively.

• UPML extends over 4, 1, .25 MRTD scaling cells respectively.

• Excellent agreement between FDTD and interface results is observed.


Dipole Radiation : Results (cont-d)

Time Steps = 20, 100, 200, 600 are shown. 3 by 3 order MRTD mesh (2x2 scaling cells).


6 grid point PML (0.1875 of a cell )

Characterization of a dielectric slab waveguide (4th order MRTD)

Scaling cell size = 8 mm (=1.28min).

5 wavelet levels to bring the effective cell size down to min /25 (range : 0-30 GHz).

The whole slab is included in one cell.

0 5 10 15 20 25 30-70

-60

-50

-40

-30

-20

-10

0

Frequency [GHz]

S11 [dB

s]

MRTD TL theory

0 5 10 15 20 25 30-70

-60

-50

-40

-30

-20

-10

0

Frequency [GHz]

S11 [dB

s]

MRTD TL theory


Wavelet Based Adaptivity : Nonlinear Wave Propagation

• As an example, the discretization of a system of equations and its boundary conditions describing a pulse compression in an optical fiber filter is presented.

• Fiber index of refraction is field intensity – dependent, assuming the form :

• Co - sinusoidal term is due to a grating written within the core of the fiber.

• Pulse compression comes as a result of negative dispersion that causes the rear of the pulse to travel faster than the front of it.

• Numerical modeling via an S-MRTD scheme (Battle-Lemarie scaling functions) was pursued in [Krumpholz and Katehi, MTT-T, 1997]. Larger – than – FDTD execution time was reported (factor of 1.5), despite the application of coarser grids.

References : H. Winful, Appl. Phys. Lett., Mar. 1986

Sarris and Katehi, 2002 Proc. IEEE IMS

2

2010 2cos)( Enznnzn


Nonlinear Wave Propagation : Results

Results from Haar MRTD (with no adaptivity) are

compared to FDTD and excellent agreement is

observed.

In terms of execution time, MRTD appears to be

slower than FDTD by ~15%.

This slowdown motivates our next step towards

formulating adaptive MRTD schemes.

Parameter FDTD, I MRTD, I FDTD, II MRTD, II

Cells 1000 500 500 250

CPU time [sec] 43 48 22 24

Max|EF(z=L)|2 10.6042 10.6410 10.3541 10.4281


is the threshold under which a wavelet term contribution to the total field is insignificant.

2/2, rjn

prE jnjn EE pr ,

times the scaling value of the cell is the (dynamic) threshold under which a wavelet

term contribution to the total field is insignificant.

• Both conditions are implemented as if – tests in the code.

• Hard thresholding is applied in this work (simpler to implement, works well for

wave propagation problems).

Application of Thresholding in MRTD

Absolute Thresholding Condition (“hard” thresholding)

Two Approaches

Relative Thresholding Condition (“soft” thresholding)


Adaptive Meshing in MRTD : Front Tracking

Conventional thresholding approaches are based on testing of thresholding conditions throughout the mesh.

This results in significant operationsignificant operation overheadoverhead.

Instead, one can track the track the wavefrontswavefronts propagating in a certain geometry and apply the tests only within those.

Example : 0 – 10 GHz pulse incidence on a dielectric slab of permittivity 2.2 (dots indicate boundaries of non-thresholded coefficients).


Operation Savings in Adaptive MRTD : Algorithm

Initialize all coefficients as “active”

Use only active wavelet coefficients

Apply update equations for scaling terms

Apply update equations for “active” wavelet terms

Determine which active wavelet coefficients remain above the hard threshold.

Apply thresholding every N time steps

Designate as active all nearest neighbors of active wavelet terms (pivot elements).

Front tracking step : active wavelet coefficients

: pivot elements

Reference : Sarris and Katehi, Proc. 2002 IEEE IMS, long paper to be submitted to MTT-T, Aug. 2002.


Execution Time Savings : Linear Wave Propagation

Timing measurements for the 0-10 GHz

pulse propagation were performed for

FDTD and adaptive MRTD.

For thresholding, the previous algorithm

is applied at each time step.

As the domain grows large, MRTD

becomes twicetwice as fast as FDTD.


Front Tracking : Nonlinear Wave Propagation

Front evolution along the characteristic line z = ct is observed.

Wavelets track the nonlinear evolution of the wavefront, assuming higher values as the

pulse gets compressed (implying higher spatial field derivatives around the front).


Comparison to FDTD for the example of optical pulse compression

Nonlinear Wave Propagation : Execution Time Savings

Thresholding window stands for the number of time steps between two subsequent applications of thresholding to wavelet coefficients.

For stability reasons, if s is the CFL number, the maximum window should not exceed 1/s.

Different thresholding windows are compared and shown to yield similar CPU time performance.

Assuming stable performance, sparser thresholding checks imply smaller operation economy.


Nonlinear Wave Propagation : Thresholding Induced Error

Using the unthresholded MRTD code as a reference, the relative error in the peak intensity at the end of the fiber filter is determined.

CPU time is reduced compared to FDTD by a factor of 30%, with absolute errors limited to the order of 0.1 %.

This case study involves complex operations in a domain largely occupied by the pulse.

Rel

ativ

e E

rror

in P

eak

Inte

nsity

[%]


0t/t40

30y et/tt/t4

zx

1tJ

00 f4

1t

GHz1f0

mm5.2zx

secp5.4t

Adaptive Meshing and Parallelization : Dipole Radiation 1

• A critical issue for the performance of parallel codes is load balancing along multiple processors.

• Adaptive repartitioning of a certain domain can be based on using wavelet coefficients of cells, as load measures (weights).

• Domain repartitioning assigns cells with large wavelet coefficients to more processors and distributes inactive cells to the rest.

• Case study : Hertz dipole radiation.

256

256

1 Joint work with P. Czarnul, University of Michigan, Prof. K. Tomko, University of Cincinnati.


Adaptive Meshing and Parallelization : Dipole Radiation

Zero order Haar MRTD, absolute threshold for wavelet coefficients = 1.0

Unthresholded wavelet coefficients (mesh refinement by 4)

Normalized Ey Field Plot

• Parallelization on 64 processors.

• Repartitioning of the domain every 500 time steps (based on Zoltan software).

• Parallelization on 64 processors.

• Repartitioning of the domain every 500 time steps (based on Zoltan software).


Cosite Interference in Ad-Hoc Networks : Problem statement

Mobile transmit - receive modules on army vehicles in proximity of each other, may corrupt each other’s performance (desensitization, intermodulation, cross-modulation etc.).

Mobile transmit - receive modules on army vehicles in proximity of each other, may corrupt each other’s performance (desensitization, intermodulation, cross-modulation etc.).

Vehicle #1Vehicle #1

Modeling Approach : Unified treatment of electromagnetic propagation across the physical channel and the operation of front – end electronics (including amplifier nonlinearities ).

Vehicle #2Vehicle #2

Mutually coupled

monopoles

T/R T/R switchswitch

A

nt e

nna

Ante

nna

Low Noise AmplifierLow Noise Amplifier

MixinMixing g

stagestagess

Power AmplifierPower Amplifier

Courtesy : humvee.net


Receiver Modeling : Lumped Elements in Time-Domain Schemes

For the modeling of lumped element loading of the antennas, the state equation approach of [1] is implemented.

This technique allows for the inclusion of active/nonlinear loads as parts of the input stage of the transceiver.

[1] : B. Houshmand, T. Itoh, M. Picket-May : “High-Speed Electronic Circuits with Active and Non-linear Components”, ch. 8 in Advances in Computational Electrodynamics : The FDTD Method, A. Taflove, ed.

sdJ

ldH

sdDdt

d

Device current

Displacement current

Total input current

+


Receiver Modeling : Lumped Elements in FDTD (cont-d)

Electromagnetic wave / device interaction is described by Norton/Thevenin type equivalent circuits (Norton equivalent is shown here).

Idev

)()()( tVdt

dCtItI devNdevN

The previous form of Ampere’s law at the device port can be interpreted as Kirchhoff’s current law for an equivalent circuit :

xzyCN /0 Equivalent FDTD cell capacitance


Equivalent circuit for an analog VHF receiver

Equivalent circuit for Yee-cell

2 section Chebychev

band-pass filter

Impedance matching

Impedance matching

MESFET-transistor

Single balanced mixer

3 section Chebychev low-

pass filter

Large signal model

(amplifier operation in the saturation region is included).

Large signal model

(amplifier operation in the saturation region is included).


• MESFET-amplifier gain for a 50MHz input signal with a power level of –10dbm, under cositing conditions is simulated in HP ADS. • Loss in sensitivity up to 25 dB is observed due to interference-driven saturation of the amplifier.

Gain without an interfering signal

Gain with an interfering signal

with a power level of 10 dBm

Frequency of the interfering signal

Gain with an interfering signal

with a power level of 13 dBm

Cositing Effect on Receiver Performance


• Monopole antennas on a PEC• Antenna length: 1.2m• Antenna impedance: 36 Ohm• Distance between antennas: 1.5m

50 Ohm source:• available power: -20dBm• frequency: 50MHz

MESFET receiver: • tuned to 50MHz• input impedance: 49 Ohm

IF-frequency: 1MHzIF-frequency: 1MHz

Simple Cosite Interference Scenario : Receiver Sensitivity

• IF power under no interference is –30.7 dBm.• IF power under no interference is –30.7 dBm.


50 Ohm source:• available power: -20dBm• frequency: 50MHz

MESFET receiver: • tuned to 50MHz• input impedance: 49 Ohm

50 Ohm source:• available power: 40dBm• frequency: 51MHz

Loss in sensitivity of about 21.7 dB

Simple Cosite Interference Scenario : Receiver Sensitivity (cont-d)

• The previous case is repeated in the presence of a parasitic transmitter.

• IF power under interference is –52.4 dBm.

• IF power under interference is –52.4 dBm.

1.5 m

1.5 m


F = 35 MHzF = 35 MHz F = 75 MHzF = 75 MHz

Dipole gap Dipole gap

Platform Modeling

• A single FDTD run yields field distributions around the platform at several frequencies, via an on-the-fly Fourier transform of time-domain data.

• LO power distribution (in dBm) around a vehicular transceiver (area = 7.5 m x 10.2 m, -40 dBm available LO power is assumed at the antenna gap).


Effect of Platform : Realistic vehicle model

Magnitude of the electric field – cross section at the plane of the monopole

gap.

Surfaces indicate nodes with a constant electric

field (at 50 MHz)

Hummer vehicle (viewpoint.com), cell size = 5cm x 5cm x 5cm


Characterization of propagation paths

Frequency: 50 MHzLength of antenna: 1.2mAntenna impedance: 28 Ohm

Frequency: 50 MHzLength of antenna: 1.2mAntenna impedance: 28 Ohm

• Cosite interference power levels at a multi-antenna system can be determined from time domain simulations, for multiple frequencies.

• In the following scenario, one antenna is excited by a broadband Gaussian pulse.

• Input power levels at the passive antennas characterize the propagation paths that determine the extent of cositing effects in the network.

-23.7dB

-10.3dB

1.5 m

1.5 m


Problem Statement

The in-forest communication between multi-antenna mobile transmit-receive units is considered. Issues to address :

• Forest propagation and multi-path (FDTD modeling requires enormous resources) .

• Effect of arbitrary platform (MoM requires extremely complex Green’s function).

• Operation of transceiver electronics under cosite interference conditions (MoM incompatible with SPICE type solvers as TRANSIM).

Modeling Approach

• Use the Method of Moments [Sarabandi and Koh, IEEE AP-49, Feb. 2001] to model wave propagation through the forest.

• Enclose the vehicular transceivers in an FDTD mesh to model rigorously the effect of the platform and the transceiver architecture as in the previous examples.

T/R switch

Ante

nna

Low Noise Amplifier

Mixing

stages

Power Amplifier

Cosite Interference in a Forest Environment


Forest propagation of a transmitted signal is modeled by MoM for multiple frequency points in a band of

interest.

Time domain waveform of the incident

(transmitted) field on the rectangular FDTD

domain is extracted via Fourier transform.

Time domain MoM data are interpolated to provide an FDTD

excitation signal at the FDTD time step.

FDTD simulation of the rectangular domain

including the vehicular transceivers is then

executed.

INHERENT ASSUMPTION : Reflected waves from the FDTD domain towards the forest and back have a secondary effect on the solution.

FUTURE WORK : Take this effect into account by coupling the two codes.

FDTD / MoM Connection Algorithm


• Validation case : A 2x2x2 FDTD mesh is used to model free space within the forest (in the absence of any vehicles). The same problem is solved by a pure MoM code.

• Results of the two methods are in excellent agreement.

(y)

(z)(x)

2x2x2 FDTD mesh and electric field component that is plotted in the figure on the right.

FDTD simulation conditions :

x =y = z = 0.3 m

t = 0.314 nsec

FDTD / MoM Connection Algorithm : Validation


The following two cases are simulated and compared :

• One antenna (with input filter tuned at 49 MHz) receives while its neighbor transmits a 1-V amplitude tone at 50 MHz.

• The amplitude of the sinusoidal transmitter is set to zero.

In both cases a transmitted 10-100 MHz from a remote in-forest transmitter excites the domain.

0.9m

Sinusoidal

transmitter

Receiver communicating with an in-forest transmitter

Input voltage at the receiving antenna

Cosite Interference in a Forest : Results


• The IF current in the two previous cases is derived and plotted in the time and frequency domain (based on 50,000 time steps).

• In the presence of the interfering signal, a 2 MHz IF corresponding to the detection of the 50 MHz tone dominates the IF spectrum.

• Loss of receiver sensitivity is observed through the drop (up to 15 dB) in the IF power level between 0-1.8 MHz ( IF filter 3-dB bandwidth is 2 MHz ).

• The IF current in the two previous cases is derived and plotted in the time and frequency domain (based on 50,000 time steps).

• In the presence of the interfering signal, a 2 MHz IF corresponding to the detection of the 50 MHz tone dominates the IF spectrum.

• Loss of receiver sensitivity is observed through the drop (up to 15 dB) in the IF power level between 0-1.8 MHz ( IF filter 3-dB bandwidth is 2 MHz ).

Cosite Interference in a Forest : Results (cont-d)


The road ahead : Coupled Problems in Electromagnetics

HTσt

Tρcp

HTσ

t

Tρcp

Source term

from EM-simulation

Source term from EM-simulation

EjH

Temperaturenode in theMEMS membrane

kjikji

zkji

ykji

xkji VEEEH ,,

2

,,

2

,,

2

,,,,

ˆˆˆ2

1

kji

kjiz

kjiy

kjixkji VEEEH ,,

2

,,

2

,,

2

,,,,

ˆˆˆ2

1

Down position : Hot spot on the bridge

Up position : Overall heating of the membrane

Acknowledgement : Kelly Tornquist, Dr. Werner Thiel, University of Michigan

Global modeling : Joint modeling of thermal, mechanical, electromagnetic effects in RF circuits.

Examples of interest : Thermal effect – induced failures in MEMS switches and multi-element

circuit boards.


The road ahead : Fast Circuit Modeling and Macro-modeling

Time – Domain characterization of complex circuit

structures provides the first step for the

implementation of reduced order modeling of those.

Equivalent Circuit Generation is based on the

extraction of time domain scattering signals, by

means of TD full wave analysis.

Systematic approaches to model extraction based on

canonical representation of multi-port network

admittance matrices (e.g. Foster representation) can

lead to automation of the process.

Reference : Mangold and Russer, IEEE MTT-T, June 1999.


The road ahead : Propagation and Wireless Channel Modeling Time Domain diakoptics (based on T-D Green’s functions), adaptive meshing, parallel computing

make time-domain modeling of propagation problems a rigorous/feasible alternative to approximate

and moment-method based techniques.

Impedance

Acknowledgement : Dr. Frank Ryan, SPAWAR, San Diego, Prof. Kamal Sarabandi, University of Michigan

15.5 m


The road ahead : Electromagnetic Modeling for State-of-the-Art Applications

Time / Frequency Domain numerical techniques and analytical methods offer insight to several

cutting-edge research areas :

Biomedical engineering (tumor detection and classification via imaging techniques).

Characterization of materials, meta-materials and random media (Sommerfeld problem, full-wave numerical modeling).

Large scale digital circuit design / signal integrity analysis.

Extensions of this work can enrich research on novel modeling techniques :

Adaptive Mesh Refinement techniques for electromagnetic propagation.

Stiff problem modeling (computational chemistry) for coupled problems.

Vector multi-resolution techniques in time/frequency domain and hierarchical finite elements in the time domain.

Time-Frequency distributions.

Time-Domain Electromagnetic Analysis : From FDTD to Wavelets and Beyond

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